A gym charges its members a onetime $36 enrollment fee and a membership fee of $19 per month. If there...

1. TRANSLATE the problem information

• Given information:
  • One-time enrollment fee: \(\$36\)
  • Monthly membership fee: \(\$19\) per month
  • Total amount charged: \(\$188\)
  • Find: number of months
• What this tells us: We need to find when one-time cost + monthly costs = total cost

2. INFER the mathematical relationship

• The total cost consists of two parts:
  • Fixed one-time enrollment fee (\(\$36\))
  • Variable monthly fees (\(\$19\) × number of months)
• This gives us the equation: \(\$36 + \$19\mathrm{x} = \$188\)
  (where x = number of months)

3. SIMPLIFY by solving the linear equation

• Subtract the enrollment fee from both sides:
  \(\$188 - \$36 = \$19\mathrm{x}\)
  \(\$152 = \$19\mathrm{x}\)
• Divide both sides by 19:
  \(\mathrm{x} = \$152 \div \$19 = 8\)

Answer: C. 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret the problem structure and forget to include the one-time enrollment fee in their equation, setting up \(\$19\mathrm{x} = \$188\) instead of \(\$36 + \$19\mathrm{x} = \$188\).

This leads to \(\mathrm{x} = \$188 \div \$19 \approx 9.9\), which doesn't match any answer choice exactly, causing confusion and potentially leading them to select Choice D (10) as the closest value.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equation correctly but make arithmetic errors, such as incorrectly calculating \(\$188 - \$36 = \$142\) instead of \(\$152\), or miscalculating the final division.

This may lead them to select Choice A (4) or Choice B (5) depending on the specific calculation error made.

The Bottom Line:

This problem tests whether students can correctly model a real-world situation with both fixed and variable costs, then execute the algebra accurately. The key insight is recognizing that total cost = one-time fee + (monthly fee × time period).